arXiv subjects
Publications and source records attributed to David A. Jorgensen.
We characterize all first cosyzygy modules of a given finitely generated module over a local ring. We do this by introducing the notion of a canonical cosyzygy module, and showing how other cosyzygy modules can be obtained from this module. As an application, we characterize Gorenstein rings via properties of canonical first cosyzygy modules.
To every local complete intersection ring one may associate a so-called generic hypersurface. In this paper we introduce rank varieties for modules and complexes over the generic hypersurface. The definition uses extension of scalars, rather than restriction of scalars which are used to define the conventional support varieties over a local complete intersection. We show that every projective variety can be realized as the rank variety of a finitely generated module over the generic hypersurface. We also investigate several properties of these rank varieties.
Given a graded-commutative ring acting centrally on a triangulated category, our main result shows that if cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.
We lay out the theory of a multiplicity in the setting of a triangulated category having a central ring action from a graded-commutative ring $R$, in other words, an $R$-linear triangulated category. The invariant we consider is modelled on those for graded modules over a commutative graded ring. We show that this invariant is determined by the leading coefficients of the Hilbert polynomials expressing the lengths of certain Hom sets. Our theory is a natural analogue of Hochster's theta invariant for homology and Buchweitz's Herbrand difference for cohomology. Moreover, we give applications to vanishing of cohomology and modules over local complete intersection rings, group algebras of a finite group, and certain finite dimensional algebras.
For a pair of finitely generated modules $M$ and $N$ over a codimension $c$ complete intersection ring $R$ with $\ell(M\otimes_RN)$ finite, we pay special attention to the inequality $\dim M+\dim N \leq \dim R +c$. In particular, we develop an extension of Hochster's theta invariant whose nonvanishing detects equality. In addition, we consider a parallel theory where dimension and codimension are replaced by depth and complexity, respectively.
Let $M$ and $N$ be finitely generated graded modules over a graded complete intersection $R$ such that $\operatorname{Ext}_R^i(M,N)$ has finite length for all $i\gg 0$. We show that the even and odd Hilbert polynomials, which give the lengths of $\operatorname{Ext}^i_R(M,N)$ for all large even $i$ and all large odd $i$, have the same degree and leading coefficient whenever the highest degree of these polynomials is at least the dimension of $M$ or $N$. Refinements of this result are given when $R$ is regular in small codimensions.
In this article we investigate a pair of surjective local ring maps $S_1\leftarrow R\to S_2$ and their relation to the canonical projection $R\to S_1\otimes_R S_2$, where $S_1,S_2$ are Tor-independent over $R$. Our main result asserts a structural connection between the homotopy Lie algebra of $S:=S_1\otimes_R S_2$, denoted $π(S)$, in terms of those of $R,S_1$ and $S_2$. Namely, $π(S)$ is the pullback of (adjusted) Lie algebras along the maps $π(S_i)\to π(R)$ in various cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on André-Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincaré series of the common residue field of $R,S_1,S_2$ and $S$.
In this paper we give a purely categorical construction of d-fold matrix factorizations of a natural transformation, for any even integer d. This recovers the classical definition of those for regular elements in commutative rings due to Eisenbud. We explore some natural functors between associated triangulated categories, and show that when d=2 these are full and faithful, and in some cases equivalences.
In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the following. (1) Over a quotient of a regular local ring by a regular sequence, every finitely generated module has finite quasi-projective dimension. (2) The Auslander--Buchsbaum formula and the depth formula for modules of finite projective dimension remain valid for modules of finite quasi-projective dimension. (3) Several results on vanishing of Tor and Ext hold for modules of finite quasi-projective dimension.
We initiate the study of modules of constant Jordan type for quantum complete intersections, and prove a range of basic properties. We then show that for these algebras, constant Jordan type is an invariant of Auslander-Reiten components. Finally, we classify modules with stable constant Jordan type [1] or [n-1] in the 2-generator case.
Let $(Q,\mathfrak n,k)$ be a commutative local Noetherian ring, $f_1,\dots, f_c$ a $Q$-regular sequence in $\mathfrak n$, and $R=Q/(f_1,\dots,f_c)$. Given a complex of finitely generated free $R$-modules, we give a construction of a complex of finitely generated free $Q$-modules having the same homology. A key application is when the original complex is an $R$-free resolution of a finitely generated $R$-module. In this case our construction is a sort of converse to a construction of Eisenbud-Shamash yielding a free resolution of an $R$-module $M$ over $R$ given one over $Q$.
We prove a generalized Dade's Lemma for quotients of local rings by ideals generated by regular sequences. That is, given a pair of finitely generated modules over such a ring with algebraically closed residue field, we prove a sufficient (and necessary) condition for the vanishing of all higher Ext or Tor of the modules. This condition involves the vanishing of all higher Ext or Tor of the modules over all quotients by a minimal generator of the ideal generated by the regular sequence.
Let $R$ be a commutative local ring. We study the subcategory of the homotopy category of $R$-complexes consisting of the totally acyclic $R$-complexes. In particular, in the context where $Q\to R$ is a surjective local ring homomorphism such that $R$ has finite projective dimension over $Q$, we define an adjoint pair of functors between the homotopy category of totally acyclic $R$-complexes and that of $Q$-complexes, which are analogous to the classical adjoint pair between the module categories of $R$ and $Q$. We give detailed proofs of the adjunction in terms of the unit and counit. As a consequence, one obtains a precise notion of approximations of totally acyclic $R$-complexes by totally acyclic $Q$-complexes.
We introduce higher-order support varieties for pairs of modules over a commutative local complete intersection ring, and give a complete description of which varieties occur as such support varieties. In the context of a group algebra of a finite elementary abelian group, we also prove a higher-order Avrunin-Scott-type theorem, linking higher-order support varieties and higher-order rank varieties for pairs of modules.
Using an alternate description of support varieties of pairs of modules over a complete intersection, we give several new applications of such varieties, including results for support varieties of intermediate complete intersections. Simpler proofs of known results are also given.
We prove that one can realize certain triangulated subcategories of the singularity category of a complete intersection as homotopy categories of matrix factorizations. Moreover, we prove that for any commutative ring and non-zerodivisor, the homotopy category of matrix factorizations embeds into the homotopy category of totally acyclic complexes of finitely generated projective modules over the factor ring.
We consider the homotopy category of complexes of projective modules over a Noetherian ring. Truncation at degree zero induces a fully faithful triangle functor from the totally acyclic complexes to the stable derived category. We show that if the ring is either Artin or commutative Noetherian local, then the functor is dense if and only if the ring is Gorenstein. Motivated by this, we define the Gorenstein defect category of the ring, a category which in some sense measures how far the ring is from being Gorenstein.
Auslander's depth formula for pairs of Tor-independent modules over a regular local ring, depth(M \otimes N) = depth(M) + depth(N) - depth(R), has been generalized in several directions over a span of four decades. In this paper we establish a depth formula that holds for every pair of Tate Tor-independent modules over a Gorenstein local ring. It subsumes previous eneralizations of Auslander's formula and yields exact bounds for vanishing of cohomology over certain Gorenstein rings.